Question

find the value of the limit for the sequence given
{1.9.17...(8n+1)/(8n)^2}

Answer 1

@jim_thompson5910 sorry to bother you again but for this problem we just focus on (8n+1)/(8n)^2 right?

Answer 2

why are there multiple decimal points in 1.9.17

Answer 3

that's just confusing notation

Answer 4

are they supposed to be commas?

Answer 5

idk no they are points that is how my teacher gave us our homework

Answer 6

are you able to take a screenshot and show me the full problem?

Answer 7

yeah i'll do that

Answer 8

here it is

Answer 9

ok let me look it over

Answer 10

ok let's say n = 1. What is the value of that expression you gave in the screenshot?

Answer 11

you can give an approximation

Answer 12

1.25

Answer 13

my bad its .1406

Answer 14

8n+1 = 9 when n = 1

so the numerator will have 1*9 = 9

the denominator is equal to (8n)^2 = (8*1)^2 = 64

So the expression is equal to 9/64 = 0.140625

Answer 15

yes

Answer 16

when n = 2, what is the expression equal to?

Answer 17

.0664

Answer 18

its approaching zero isnt it?

Answer 19

8n+1 = 17 when n = 2

so the numerator will have 1*9*17 = 153

the denominator is equal to (8n)^2 = (8*2)^2 = 256

So the expression is equal to 153/256 = 0.59765625

Answer 20

the numerator is NOT simply 17

it's 17 times the previous values (1 and 9)

Answer 21

when n = 3, the numerator isn't simply 25

it is 25*17*9*1 = 3825

Answer 22

oh so we have to multiply times the other three values

Answer 23

the numerator is basically the product function

\[\Large \prod_{k=0}^{n} (8n+1) \]

Answer 24

The big upper case pi stands for product

Answer 25

It's analogous to the sigma sum function

Answer 26

oops typo, meant to write

\[\Large \prod_{k=0}^{n} (8k+1)\]

Answer 27

\[\Large \prod_{k=0}^{1} (8k+1) = 1*9\]
\[\Large \prod_{k=0}^{2} (8k+1) = 1*9*17\]
\[\Large \prod_{k=0}^{3} (8k+1) = 1*9*17*25\]
...
...
...
\[\Large \prod_{k=0}^{n} (8k+1) = 1*9*17*25*\ldots*(8n+1)\]

Answer 28

oh so we are just taking the limit of the denominator?

Answer 29

or the whole thing

Answer 30

we're doing the whole thing. It's a bit tedious, so I had excel do much of the work

check out the attached document

Answer 31

notice at n = 26, we have the whole expression equal to 4.51 * 10^45 which is a very huge number

Answer 32

oh man did you create that chart?

Answer 33

those are some pretty huge numbers man

Answer 34

I used the computer to do much of the work

Answer 35

thats pretty cool man

Answer 36

so the value for this limit problem is infinity

Answer 37

yes